You are good with Y or you want Y? I see July 3rd has Y HND-ORD. You are correct that right now Y is scarce in that time frame, but I do see a bunch of Premium Econ. Y is 45k and PE is 55k. Sounds like a deal to me.

Edited to add:
If you really want Y and you are doing a round-trip, say ORD-HND, HND-ORD, you could pair an outbound JL Econ with a Korean Y award the other direction (and potentially swap it out with JL if JL becomes available later). Korean Y awards show up as 70k one-way because KE requires a round-trip to get lowest pricing. If you book a KE Y award as part of a round-trip it prices at 35k, cheaper than JAL (but requires a connection in Seoul). You can pair JL Y one direction with KE Y the other which gives you the 35k rate on the KE Y (you do not have to fly KE both directions, you only need to have a round-trip, any other carrier can pair with KE to get the lower round-trip award pricing on KE).

The small caveat being that I believe there are no changes allowed to the return once you fly the outbound so if JL opens up after you fly the outbound of a round-trip you would not be able to change from KE to JL on the return.